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Constructive Approaches to the Rigidity of Frameworks Viet Hang Nguyen

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Constructive Approaches to the Rigidity of Frameworks Viet Hang Nguyen Constructive approaches to the rigidity of frameworks Viet Hang Nguyen To cite this version: Viet Hang Nguyen. Constructive approaches to the rigidity of frameworks. Math´ematiques g´en´erales[math.GM]. Universit´ede Grenoble, 2013. Fran¸cais. <NNT : 2013GRENM052>. <tel-00992387> HAL Id: tel-00992387 https://tel.archives-ouvertes.fr/tel-00992387 Submitted on 17 May 2014 HAL is a multi-disciplinary open access L'archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destin´eeau d´ep^otet `ala diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publi´esou non, lished or not. The documents may come from ´emanant des ´etablissements d'enseignement et de teaching and research institutions in France or recherche fran¸caisou ´etrangers,des laboratoires abroad, or from public or private research centers. publics ou priv´es. THÈSE Pour obtenir le grade de DOCTEUR DE L’UNIVERSITÉ DE GRENOBLE Spécialité : Mathématique et Informatique Arrêté ministériel : 7 aoutˆ 2006 Présentée par Việt Hằng Nguyễn Thèse dirigée par Zoltán Szigeti préparée au sein du Laboratoire G-SCOP et de l’École Doctorale MSTII Constructive approaches to the rigidity of frameworks Thèse soutenue publiquement le 17 octobre 2013, devant le jury composé de : M. Jorge Ramírez Alfonsín Professeur, Université Montpellier 2, Président M. András Recski Professeur, Budapest University of Technology and Economics, Rapporteur M. Walter Whiteley Professeur, York University (Toronto), Rapporteur M. Victor Chepoi Professeur, Université de la Mediterranée (Marseille), Examinateur M. Tibor Jordán Professeur, Eotv¨ os¨ Loránd University (Budapest), Examinateur M. Zoltán Szigeti Professeur, Grenoble INP, Laboratoire G-SCOP, Directeur de thèse Acknowledgements I would like to thank the members of the committee for their comments, suggestions on the manuscript as well as inspiring questions. I would like to express my deepest thank to my advisor Zolt´an Szigeti for his constant support, encouragement, counsel and especially the great freedom I enjoyed during these three years. My special thank goes to Andr´as Seb¨ofor his support and also for his energy which propagates in our research group and motivates us all. I am deeply grateful to Olivier Durand de Gevigney, my academic brother, co-author and true friend. I greatly appreciate his help and all the time we have shared. I would like also to thank my other collaborators Abdo Alfakih, Bill Jackson and Tibor Jord´an for the joint works and many things I learned from them. I am thankful to all the people in our research group for discussions and sem- inars. I thank friends and staffs in the G-SCOP laboratory for having created a pleasant atmosphere. I would like to express my thank to my family in Vietnam for their love and encouragement. I am deeply grateful to my family-in-law and especially to my mother-in-law for her kindness and help. There are no words to express my thank to my husband Guillaume who has been and will be always with me in our journey of life. I am also thankful to our little angel Daniel Hai Phong, our unending source of joy and inspiration. 1 2 Contents 1 Introduction 7 1.1 Frameworks and rigidity ........................ 10 1.2 Our contributions and organization of the thesis ........... 15 2 Preliminaries 19 2.1 Basic notions .............................. 20 2.2 Graphs and digraphs .......................... 20 2.3 Matroid theory ............................. 23 2.4 Submodular functions ......................... 26 2.5 Algebra and linear algebra ....................... 27 2.6 Congruent motions and motions of rigid bodies ........... 30 3 Rigidity theory 33 3.1 Introduction ............................... 34 3.2 Various types of rigidity ........................ 35 3.2.1 Local rigidity .......................... 35 3.2.2 Infinitesimal rigidity and rigidity matrix ........... 36 3.2.3 Local rigidity versus infinitesimal rigidity ........... 38 3.2.4 Static rigidity .......................... 40 3.2.5 Infinitesimal rigidity versus static rigidity ........... 41 3.2.6 Global rigidity .......................... 41 3.2.7 Universal rigidity ........................ 42 3.2.8 Stress matrices ......................... 42 3.3 Combinatorial rigidity ......................... 44 3 Contents 4 Matroid approach 51 4.1 Introduction ............................... 52 4.2 Characterizing abstract rigidity matroids ............... 53 4.3 1-extendable abstract rigidity matroids ................ 54 4.4 Submodular functions inducing ARMs ................ 58 4.5 A potential application ......................... 61 5 Inductive constructions and decompositions 63 5.1 Introduction ............................... 65 5.2 Graded sparse graphs .......................... 68 5.2.1 Introduction ........................... 68 5.2.2 A reduction theorem for graded sparse graphs . 70 5.2.3 Inductive construction of graded sparse graphs . 75 5.2.4 Decomposition of graded sparse graphs ............ 75 5.2.5 Graded sparse matroids .................... 81 5.3 (b, l)-sparse graphs ........................... 86 5.3.1 Inductive construction ..................... 86 5.3.2 (b, l)-pebble games ....................... 93 5.4 Packing of matroid-based arborescences ................ 96 5.4.1 Introduction ........................... 96 5.4.2 Proof of the main theorem ...................102 5.4.3 Polyhedral description .....................105 5.4.4 Algorithmic aspects .......................106 5.4.5 Further remarks .........................107 6 Infinitesimal rigidity 109 6.1 Introduction ...............................110 6.2 Body-bar frameworks with boundary .................112 6.2.1 Body-bar frameworks ......................112 6.2.2 Body-bar frameworks with bar-boundary ...........115 6.3 Body-length-direction frameworks ...................119 6.3.1 Introduction ...........................119 6.3.2 Frameworks with length-direction-rigid bodies ........120 6.3.3 Frameworks with direction-rigid bodies ............122 6.3.4 Frameworks with length-rigid bodies .............126 6.4 Direction-length frameworks ......................131 6.4.1 Introduction ...........................131 4 Contents 6.4.2 0-extensions for direction-length graphs ............133 6.4.3 1-extensions for direction-length graphs ............135 6.4.4 Union of two spanning trees ..................141 7 Global rigidity of direction-length frameworks 143 7.1 Introduction ...............................144 7.2 Quasi-generic direction-length frameworks ..............145 7.3 Proof of Theorem 7.1.2 .........................147 7.4 Further remarks .............................156 8 Universal rigidity 159 8.1 Introduction ...............................160 8.2 Universal rigidity in R1 .........................161 8.2.1 Universal rigidity of bipartite frameworks ...........161 8.2.2 Bipartite graphs in general dimension .............164 8.2.3 Observations, open questions, conjectures ...........165 8.3 Universal rigidity of tensegrity frameworks ..............172 8.3.1 Introduction ...........................172 8.3.2 Preliminaries ..........................175 8.3.3 The configuration matrix and Gale matrices .........177 8.3.4 Dominated tensegrity frameworks ...............178 8.3.5 Affinely dominated tensegrity frameworks ..........180 8.3.6 Proof of main results ......................183 8.3.7 Counter-example and conjectures ...............185 Conclusion 189 References ...................................193 5 Contents 6 Chapter 1 Introduction Contents 1.1 Frameworks and rigidity .................. 10 1.2 Our contributions and organization of the thesis . 15 7 La th´eorie de la rigidit´ese pr´eoccupe originellement de la rigidit´e/flexibilit´edes charpentes. Une charpente est un mod`ele math´ematique d´ecrivant une structure r´eelle. Une des premi`eres sources de structures qui ont suscit´el’´etude de la rigidit´e vient de l’ing´enierie des structures. Consid´erons une structure planaire constitu´ee de barres solides jointes aux extr´emit´es par des jonctions permettant `adeux barres jointes de bouger librement dans le plan autour du joint. Une question naturelle est de savoir si la structure peut ˆetre d´eform´ee. Par exemple, la structure planaire rectangulaire de la Figure 1.1 peut ˆetre d´eform´ee continˆument en une structure planaire non-rectangulaire comme repr´esent´epar la figure en pointill´es. Nous dis- ons que la structure est flexible. Au contraire, la structure planaire de la Figure 1.2 ne peut pas ˆetre d´eform´ee continˆument en une autre forme sans ˆetre cass´ee. Nous disons que la structure est rigide, ou plus pr´ecis´ement, localement rigide. Ce type de structures barres-et-joints planaires peut ˆetre mod´elis´ee par une paire (G, p) form´ee d’un graphe G = (V, E) et d’une application p : V → R2. Les sommets V de G correspondent aux joints et les arˆetes E de G correspondent aux barres. L’application p d´etermine la position des joints. De la mˆeme mani`ere, une structure barres-et-joints en dimension 3 (et plus g´en´eralement en dimension d) peut ˆetre mod´elis´ee par une application p : V → R3 (et p : V → Rd, respective- ment). Nous appelons cette paire une charpente barres-et-joints. Un d´eplacement continu d’une charpente (G, p) dans Rd est un d´eplacement continu des sommets de G dans Rd qui conserve les longueurs des arˆetes. Une charpente est dite locale- ment rigide si tous les d´eplacements continus de (G, p) pr´eservent aussi la distance entre chaque paire de sommets de G. De mani`ere
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